Let be a harmonic function and a diffeomorphism. Is (composition) harmonic in ? I know I could compute the laplacian, but is there any other method to do it?.

You need to apply the chain rule here. Define . You need to show that where is the Laplacian operator for each . To prove this let . Thus, write where and . Now by the chain rule and . Now go on and continue to and and add them up.